Suppose that A and B are square matrices, and that AB = BA. Suppose that v is an eigenvector of A with corresponding eigenvalue A. a) Show that Bu is in E, (A). Hint: We have Au = Xv. Do something to this equation to get B into the picture. b) Now also suppose that dim(E, (A)) = 1. Explain why in this case v is also an eigenvector of B. Hint: From part (a) you know both v and Bu are in E (A). What does the condition dim(E, (A)) = 1 tell you about these two vectors?

Suppose that A and B are square matrices, and that AB = BA. Suppose that v is an eigenvector of A with corresponding eigenvalue A. a) Show that Bu is in E, (A). Hint: We have Au = Xv. Do something to this equation to get B into the picture. b) Now also suppose that dim(E, (A)) = 1. Explain why in this case v is also an eigenvector of B. Hint: From part (a) you know both v and Bu are in E (A). What does the condition dim(E, (A)) = 1 tell you about these two vectors?

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 11AEXP

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